Computing Representations Using Hypervolume Scalarizations

In this article, we use the hypervolume indicator as a scalarizing function for biobjective combinatorial optimization problems. In particular, we describe a generic solution approach that determines the nondominated set of a biobjective optimization problem by solving a sequence of hypervolume scalarizations with appropriate choices of the reference point. Moreover, this solution technique can also provide a compact representation of the efficient set that is a (1-1/e)-approximation to the optimal representation in terms of the hypervolume in an a priori manner. We illustrate these concepts for a particular variant of the biobjective knapsack problem and for a biobjective shortest path problem. Numerical results are presented for the former problem.